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Full text of "Analytical Mechanics"

FIG. 21.
are arcs of circles whose planes are parallel to the plane of the paper and whose centers lie on the axis of rotation.
34.   Uniplanar Motion. When a rigid body moves so that each of its particles remains at a constant distance from a fixed plane the motion is said to be uniplanar.    The
fixed plane is called the guide plane.
35.   Theorem I.  Uniplanar motion of a rigid body consists of a succession of infinitesimal rotational displacements.
Suppose the rigid body A, Fig. 22, to describe a uniplanar motion parallel to the plane of the paper and let A and A' be any two positions occupied by the body. Then it may be brought from A to A' by a rotational displacement about an axis the position of which may be found in the following manner. Let P and Q be the positions of any two particles of the body in a plane parallel to the plane of the paper when the body is at the position A, and Pf and Qf be the positions of the same particles when the body occupies the position A'. Then the desired axis is perpendicular to the plane of the paper and passes through the point of intersection of the perpendicular bisectors of the lines PP' arid QQ', drawn in the plane determined by these lines.
Therefore the body can be brought from any position A to any other position A' by a single rotational displacement. The actual motion between A and A! will be, in general,
FIG. 22.